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This activity has been inspired by Doug Williams' Poly Plug resource. You can find out more details, including how to order sets of Poly Plug, on the Mathematics Centre website. However, you do not need sets of Poly Plug to have a go at this activity - please see below.
Why do this problem?
This challenge encourages learners to make hypotheses, devise ways to test them and then refine their thinking based on new evidence.
Display the interactivity on the board and invite a pupil to select $13$ spots. The simple fact of the score going up will hook children into the activity. Explain that their task is to work out how the scoring system works. Give them a few minutes to chat to someone else about any initial thoughts. Draw the whole group together and take a few suggestions.
Then, give pairs time to discuss what they want to do in order to find out more and test their ideas. Explain that you will want to hear (a) what they'd like to do and (b) what information this will give them. At this point, if you have a access to a computer suite, you may wish the class to work in pairs at a computer, trying out their own ideas. If this is not possible, then bring everyone together again and invite some pairs to offer their thoughts. As a whole class, come to an agreement about what you are going to try next, and give it a go. Some pupils may wish to record different arrangements and their different scores to refer back to later. This sheet of blank $5$ by $5$ grids may be useful for that purpose but children will have other methods of their own too. Continue in this way with pairs making suggestions for arrangements to test until the whole group is sure that the scoring system is understood. You could choose $13$ spots of your own and show the group your arrangement on paper or by drawing it on the board, asking them to predict the score before trying it out using the interactivity.
The challenge can then focus on obtaining the maximum possible score. Pairs could use the blank $5$ by $5$ grids to record different arrangements and make a note of their calculated scores. After a suitable length of time, you could find out who has the very high scores so far and test them on the computer.
This could be a 'simmering' activity so that children are encouraged to work on the maximum score over a period of a few days or weeks. You could create some wall space for them to post up their current top scores and pictures of the arrangements for the whole class to return to at a specified time.
What could you try next?
Why is that a good arrangement to try?
What do you think the score will be?
Some children may enjoy investigating the maximum score on a differently sized grid and possibly creating their own scoring system for someone else to decipher.
Working at a computer with a partner will give instant feedback which may help some children who might not persevere otherwise. The image in the Hint might also be useful.